K3 surfaces, rational curves, and rational points

TL;DR

This paper investigates the distribution of rational points on elliptic and K3 surfaces over number fields, establishing conditions under which points avoid rational curves, and constructs explicit examples of such points.

Contribution

It proves that on certain elliptic surfaces, points lying on finitely many rational curves imply the existence of points on no rational curves, addressing a question by Bogomolov.

Findings

Existence of algebraic points not on any rational curves on specific elliptic K3 surfaces

Conditions linking points on finitely many rational curves to points on none

Explicit construction of algebraic points avoiding all smooth rational curves

Abstract

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves.